Noise analysis to reveal jitter and crosstalk&#39;s effect on signal integrity

ABSTRACT

A method and apparatus for generating a probability density function eye are provided. The method preferably includes the steps of acquiring an input waveform, performing a clock data recovery in accordance with the input waveform to determine one or more expected transition times and defining a plurality of unit intervals of the input waveform in accordance with the one or more expected transition times. One or more values of one or more data points may then be determined in accordance with the input waveform in accordance with the one or more expected transition times, and a category for each unit interval in accordance with its state and its position within the input waveform may also be determined. One or more histograms may then be generated for the determined one or more values for each category of unit intervals.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 13/739,106, filed Jan. 11, 2013 by Miller, entitled NOISEANALYSIS TO REVEAL JITTER AND CROSSTALK'S EFFECT ON SIGNAL INTEGRITY,currently allowed, which in turn claims the benefit of i) U.S.Provisional Patent Application 61/586,341, filed Jan. 13, 2012 titled“Noise Analysis to reveal Crosstalk's effect on overall “SignalIntegrity”” to Miller; and ii) U.S. Provisional Patent Application61/586,348, filed Jan. 13, 2012 titled “Noise Analysis to revealCrosstalk's effect on overall “Signal Integrity”” to Miller, thecontents of these applications being incorporated herein by reference.

FIELD OF THE INVENTION

The invention concerns the analysis of Signal Integrity in the field ofdigital data communication and digital electronics in general.Specifically within this field, the invention concerns jitter, noise andcrosstalk analysis and is related to a method and apparatus forevaluating jitter (or timing uncertainty) and noise (voltageuncertainty) performance of a data channel or clock under observation.

BACKGROUND OF THE INVENTION

In present day, Signal Integrity analysis is concerned with jitter(timing uncertainties) and noise (voltage uncertainties) performance ofdata channel(s) or clock circuits. These two dimensions, in which theelectronic data channel or clock manifests are equally important andexcessive jitter or noise can lead to data channel malfunction. The twophenomena are actually intertwined as increased noise generally leads toincreased jitter, and jitter can result in increased noise.

This kind of analysis and investigation is generally categorized assignal integrity (SI) analysis. In recent history, most of the SI focushas been on jitter analysis. Much work has been done to devise methods(prior art) to decompose jitter into component parts which permit abetter understanding of the nature of this “uncertainty in timing” wecall jitter. Some attention has been paid to similar decomposing thenoise of the same circuits, but this area has so far beenunderexploited.

Furthermore as more and more serial data channels are packed into closeproximity the issue of “crosstalk” or unwanted interference betweencircuits has become a recognized problem to be addressed. There are anumber of ways that this unwanted interference can affect and impair theperformance, depending on the physics of the interference. One suchmechanism is electromagnetic coupling. That is the propagation of fieldsarising from the rapidly changing currents in printed circuitconductors. The principle manifestation of “crosstalk” of this kind is“noise” by nature. “Noise” may be defined as any undesired pollution ofa transmitted signal due to electronic noise (as defined in theindustry) intrinsic to a data channel's circuitry, but including anyeffects induced by neighboring active signals, whether they are otherdata channels, or simply other dynamic electronic signals (or voltagesources) in the vicinity of a data channel under observation. Some“crosstalk” from other signals is understood to be included in the“noise” which can degrade and impair a data channel, and as such isundesirable. In light of the interest in crosstalk, a closer look atnoise analysis is a logical extension of SI analysis. “Noise” in generalfor a data channel will encompass both the intrinsic noise of thechannel, as well as any perturbations induced by the aforementioned“crosstalk” from whatever other signals are in the vicinity. To furthercomplicate matters, whatever measurement instrumentation is employed to“observe” the data channel and other neighboring signals has its own“noise” contributions, and this measurement noise is as important toconsider as either the intrinsic or the “crosstalk” noise components. Tothoroughly dissect and analyze noise and whatever crosstalk may bepresent, it is important to develop a methodology that provides the mostclear characterization of which parts of the “noise” are dependent onthe average signal shape, which parts are bounded, which parts are not,and to isolate all that is not systematically related to the signalitself, so that it may be analyzed in relation to candidate crosstalksignals, for the purpose of identifying the source of the crosstalk.

The fundamental nature of an oscilloscope measurement (or waveformrecording instrument) is one that “samples” at some nominally uniformtime intervals the voltage of a signal which is presented to it. Thevoltage is a varying function over time for any data channel ofinterest, but even lacking a data channel, any voltage source hasvariations over time which are random and which are commonly known as“noise”. The sources of noise are rooted in the physics of whatevercircuit is being observed. There are many references on this subjecteasily available in text books and on the Internet.

A tool commonly used in studying noise and jitter is called an “eye”diagram. Such diagrams have been in existence for many years and offer a2 dimensional approximation of the “probability density” for the signalsunder analysis (2D eye diagrams). These 2D eye diagrams have a number ofweaknesses which are seldom discussed. One problem is that they continueto change as more and more data contributes to the eye diagram, andthere is no easy way to know when you have “enough” data. This evolutionof an eye diagram is due to the simple nature of random noise. It iswell known in statistics that the expected value of the peak-to-peak ofa Gaussian or Gaussian-like distribution of an observed set of eventsdepends on the number of events observed. As more and more events areobserved, the width of the observed distribution broadens. For example,FIG. 1 shows an eye diagram with a nine thousand UI eye [1] as it wouldlook after nine thousand unit interval (UI) have been accumulated.Furthermore, FIG. 2 shows an eye diagram with a five million UI eye [3]as it would look after five million UI have been accumulated. Asexpected the extents of the populated regions of the eye have grown asmore UI are accumulated. As such, two eye diagrams from a differentnumber of UI cannot be compared directly.

Often eye diagrams are used to perform a “mask” test, wherein a polygonor polygons are used to define regions of exclusion not to be touched bythe points in the eye diagram. The problem of course, is how much datais needed for a valid mask test, because the probability of a maskviolation depends on how many chances the signal under test is given toviolate the mask. This is fundamentally a consequence of the eye diagrambeing non-convergent. There are regions of the 5 million UI eye that areimpacted [4], whereas for the 9 thousand UI eye the same region is notimpacted [2].

One approach to solve this problem is to try to estimate from the eyediagram a “contour plot” or a 2-dimensional representation. The contourplot is a well known concept. It is supposed to represent the absoluteprobability of the signal under observation to touch a given coordinatein the eye diagram coordinates. Methods for this kind of calculationexist today. For oscilloscopes these methods suffer from ambiguity incalculating probabilities from an already formed eye. In particular inthe region of the contributions from rising edges and falling edgescontributions to the eye diagram make it impossible to know if thetrajectory of the signal under test that produced that point was earlieror later. Furthermore in an already formed eye diagram, the separationof vertical (noise) contributions from horizontal (jitter) is notpossible. For example, if one wanted to compensate the eye diagram forthe contribution of the measuring instrument's inherent noise, onecannot. Contour plots can also be generated by a Bit Error Rate Testerthat is specially equipped for this task. This same shortcoming applies,in that the noise and jitter inherent in the instrument cannot beeffectively removed from such a contour plot. It is notable that in theSI analysis prior-art there is a consortium based software tool referredto as “Stat-Eye”. This tool can produce eye diagrams based onassumptions about noise and jitter and these objects have a differentset of problems while addressing some of the defects in ordinary eyediagrams. These are essentially predictive tools dependent on electronicmodels and conscious assertions made by the user of the tool.

In general, in current SI analysis, there is no way to independentlyanalyze the spectrum on time-domain of “only” the non-deterministic partof the noise, without the spectrum of the signal itself present in thespectrum.

The inventor of the present invention has determined that both contourplots and eye diagrams would be more useful for comparing test caseswhere crosstalk is present compared to cases where crosstalk is notpresent, if the above shortcomings could be overcome. The compensationis important to minimize the impact of the measuring instrument, andimproving the quality and precision of the contour plot would be verybeneficial.

Current SI methods do permit characterization of a data pattern'ssystematic trajectory, or shape through every bit or UI of the testpattern. This is accomplished via resampling data to have exactly Nresampled points and forming a signal average from these resampledpoints. Such methods are standard in industry standardsserial-attached-SCSI (SAS) for the purpose of estimating total waveformdistortion penalty (TWDP). However these methods only supply the shapeor trajectory of the signal under test, either as a function of positionwithin a repeating sequence of test data, or as defined by thesurrounding local sequence of data states.

Therefore, the inventor of the present invention has determined thatwhat is needed is:

-   -   1. A convergent form of the eye diagram. That is one which does        not change significantly as more data is accumulated.    -   2. A means to compensate the eye diagram for the noise inherent        in the measuring instrument.    -   3. A means to overcome the inability of an oscilloscope to        produce a contour plot which extends outside the central region        of the eye.    -   4. A means to produce a contour plot which is compensated for        the inherent noise of the measuring instrument.    -   5. Good methods for visualizing effects of crosstalk.

OBJECTS OF THE INVENTION

Among others, it is an object of this invention:

-   -   1. To provide a means to produce a convergent form of the eye        diagram. That is one which doesn't change significantly as more        data is accumulated.    -   2. To provide for an eye diagram that is compensated for noise        inherent in the measuring instrument.    -   3. To overcome the inability of an oscilloscope to produce a        contour plot which extends outside the central region of the        eye.    -   4. To provide a contour plot that is compensated for the        inherent noise of the measuring instrument.    -   5. To provide methods for visualizing effects of crosstalk.

Still other objects and advantages of the invention will in part beobvious and will in part be apparent from the specification anddrawings.

SUMMARY OF THE INVENTION

One or more embodiments of the present invention may be provided toovercome the drawbacks of the prior art. To overcome the non-convergenceproblem, one or more embodiments of the invention preferably break downthe analysis of signals under observation first into unit intervals of aclock or data sequence, and then categorize each unit interval asbelonging to a category defined in such a way as to group together unitintervals that are very likely to have similar or even identicalhistory, and so may be expected to have a well defined trajectory overthe span of the unit interval. For each observed category, at severaltime intervals within and surrounding the UI (which need not beuniformly spaced) the distribution of vertical (usually voltage) valuesis preferably captured. The number and precise locations of thesedistributions is preferably chosen to adequately describe the changes invertical variations for a category, so that the vertical distributionsfor the category can be estimated over the entire breadth of a UI aswell as regions extending somewhat into adjacent UIs. By treating thecategories separately, the nature of the distributions are betterconstrained and they are much simpler to analyze and parametrize.

Once significant statistical distributions have been captured, and eachdistribution may be parametrized, and probability density maps may bemade for each category. The superposition of all categories or a subsetof categories (e.g. all categories with a transition at the start of theUI) can be superimposed to produce a convergent probability distributionfunction (PDF) map which is a superior eye diagram, the PDF-Eye. Inaddition since the parametrization of the distributions lends itself toadapting a Gaussian component of the distribution, it can be compensatedfor the measuring instrument's noise, overcoming the need for additionalcompensation.

Likewise a probability map (as distinguished from a probability density)may be created from the parametrized distributions from which a superiorcontour diagram, the cumulative distribution function (CDF)-Eye may bederived. In this way, in accordance with an embodiment of the invention,it is possible to create the probabilities of the signal under test toimpact every coordinate on the eye diagram's space, even extendingoutside the central region of the eye. Again as for the new PDF-Eye theneed to compensate for instrument noise is overcome.

Furthermore, using the same inventive strategy for categorization andaverage values rather than distributions, the creation of a systematicwaveform, with sample points at the same times as the captured inputwaveform can be formed where the non-systematic (random) variations havebeen removed. Furthermore, in accordance with an alternative embodimentof the invention, the difference between the input waveform and thesystematic waveform provides a residual waveform consisting of only thenon-systematic part of the signal under observation.

These new kinds of eye diagrams presented in accordance with one or moreembodiments of the present invention exist in several varieties asdescribed in this application, and they with the residual and systematicwaveforms provide significant means for observing jitter, noise andcrosstalk.

In accordance with additional embodiments of the invention, it isfurther possible to build an apparatus to implement the acquisition,clock recovery, capture of distributions and a processor for processingthem with a processor to produce these results and displays.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the invention, reference is made tothe following description and accompanying drawings, in which:

FIG. 1 is an eye diagram containing nine thousand UI according to priorart methods;

FIG. 2 is an eye diagram containing five million UI according to priorart methods;

FIG. 3 is the flow diagram of the three major steps in accordance withan embodiment of the preferred embodiment;

FIG. 4 is a flow diagram showing the steps of acquiring waveforms andbuilding a database of histograms in accordance with an embodiment ofthe present invention;

FIG. 5 is a flow diagram showing the steps of analysis of the histogramdatabase in accordance with an embodiment of the invention;

FIG. 6 shows samples and noise histograms across one unit interval inaccordance with an embodiment of the invention;

FIG. 7 shows samples and noise histograms across one unit interval inaccordance with an embodiment of the invention;

FIG. 8 is a noise histogram according to prior art methods;

FIG. 9 is a PDF eye diagram according to an embodiment of the presentmethod;

FIG. 10 is a contour plot containing lines of constant probabilityaccording to an embodiment of the present method;

FIG. 11 is a pair of plots containing a signal centric iso-BER plot anda signal centric contour plot in accordance with an embodiment of theinvention;

FIG. 12 is a pair of plots containing a data centric iso-BER plot and adata centric contour plot in accordance with an embodiment of theinvention;

FIG. 13 is four plots containing a data centric iso-BER plot and a datacentric contour plot along with a jitter bathtub curve and noise bathtubcurve in accordance with an embodiment of the invention;

FIG. 14 is a flat CDF eye diagram in accordance with an embodiment ofthe invention; and

FIG. 15 is a plot showing the systematic waveform and residual waveformaccording to an embodiment of the present method.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

For the purposes of this application, the procedures for acquiringdigital waveforms, subtracting them if they are differential (i.e. muchas the implicit electronic “receiver” would effectively subtract thedifferential analog counterparts), the detection of transitions foreither clock or data, clock data recovery (CDR), digital resampling andare all understood as procedures well known in the prior state of theart. As such these methods do not require further detailed descriptions.

The three basic steps of a particular embodiment of the preferredembodiments are shown in FIG. 3. Here we see a first step [5] consistingof the acquisition and analysis of waveforms to produce a database ofhistograms followed by a second step [6] consisting of the analysis ofthe database of histograms to produce a 2D PDF object [33], a 2D CDFobject [34] and a set of mean values followed by a final step [7]consisting of the generation of displays from the 2D CDF object [34], 2DPDF object [33] and the set of mean values.

Referring next to FIG. 4 in addition to FIG. 3, during this first step[5], first input waveforms may be acquired [8] and a single inputwaveform obtained [9] or [10]. Next, the input waveform is preferablyanalyzed to identify transition times [11]. A CDR procedure may then beperformed [12] resulting in the recovered times defining the start andend of each UI in the captured input waveform. Once this is accomplishedthe next step is preferably to decode the states (1 or 0) for each UI inthe waveform [13] producing a list of sequential bit states for thewaveform. Two cases must be addressed, the case where there is arepeating pattern in the sequence of states and the case where there isno such repeating pattern. Depending on which case, a categorizationmethod [14] will preferably be either: 1) the category associated witheach UI is defined by its position in the repeating pattern, or 2) ifthere is no repeating pattern, the category for a UI will be defined byan M-bit binary code consisting of the M−2 bits (or other number ofbits) prior to the UI, the state of the UI and the state of thefollowing UI (or other predetermined number or portions of UIs).

One purpose of analysis in accordance with the various embodiments ofthe invention is to uncover any non-systematic behavior, and then toisolate the systematic from the non-systematic. An essential element ofthese various elements of the invention is to determine the “average”shape of the serial data or to establish the systematic behavior.

It is well known for the purposes of establishing the shape of thetrajectory for both the repeating pattern case and the non-repeatingpattern case. For the case of a repeating pattern FIG. 6, for Nhorizontal intervals across a UI from the start of the UI [35] to theend of the UI [36], only N points [38] are needed, since the points fromthe previous and next UIs will provide the necessary points before andafter the UI in order to reproduce the trajectory of the category overthe horizontal extent of the eye. For the non-repeating case FIG. 7,more points are typically needed to establish the average trajectoryover the same horizontal extent as the traditional eye. In the preferredembodiment 2N+1 resampled data points are used (although other choicesare possible). An additional N/2 points before [41] the UI, N pointsinside [38] the UI and N/2+1 points after [42] the UI are preferablyemployed. In this preferred embodiment of the invention, instead offorming a simple average at each of the resampled points, a histogrammay be formed [40] for each of the sample points in each category fromwhich an average can later be obtained, but from which much additionalinformation can be gained about the nature of the vertical noise.

Therefore, further in accordance with one or more preferred embodimentsof the invention, in order to perform the step of analyzingsubstantially every UI in the input waveform as described above, N or2N+1 data points are interpolated from the input waveform [15] and the Nor 2N+1 histograms for the category of this UI is updated [16]. Adatabase [21] is formed, including the data point interpolated values,which is organized by the observed categories, and which consist ofeither N histograms per category, or of 2N+1 histograms per categorydepending on whether there is or is not a repeating pattern (one foreach data point included in the processing, as noted above). Multipleacquisitions may be treated in the fashion described above to accumulategood statistics and for the non-repeating pattern case to allow for rarecategories to manifest. Once an adequate amount of data points have beenacquired and interpolated, and therefore sufficient data is available toprovide meaningful statistics, the generated database of histograms canbe analyzed.

One objective of the next major step [6] is to analyze the database ofhistograms to produce a 2D PDF object [33] and 2D CDF object [34] whichcomprise inventive, novel forms of eye diagrams and contour plots. Thesewill have the same vertical and horizontal extents as would atraditional eye diagram, and so an estimate of the PDF for eachcoordinate of that area may be made. Likewise an estimate of the CDF orprobability of the signal under observation might pass through anyparticular selected coordinate may also be made.

There will therefore be generated a set of histograms for each category[21] Each histogram is fitted, which is by now a well known procedure asshown in FIG. 8, yielding 6 parameters: σ_(L), μ_(L), ρ_(L), σ_(R),μ_(R), ρ_(R). The CDF of the histogram is preferably translated to avertical Q-scale [44] according to the optimized value of ρ_(L) forwhich the data in the fit region [47] s most linear. The best fit line[45] has a slope which is the reciprocal of σ_(L) and the intercept [46]at Q=0 of that line yields the value of μ_(L). An identical proceduremay be performed on the right-hand side of the histogram to obtain ρ_(L)from optimal linearization of the region [51], to obtain σ_(R) from theslope of the line [49] and to obtain μ_(R) from the intercept [48].

For producing the average trajectory per category, the means of allhistograms per category are preferably calculated [23].

Optionally, each set of fit parameters may be modified [24] to use asomewhat smaller σ_(L) and σ_(R) reducing them by a quadraturesubtraction (σ′_(L)=√{square root over (σ_(L) ²−σ_(instrument) ²)}) of aknown random noise, σ_(instrument) contributed by the measuringinstrument.

Using these 6 parameters, each histogram can be extrapolated to a PDF[25] (i.e. including one variable, in addition to the extension alongthe time axis) using the parameters to express the low probabilitydensity values at the extremes, and simply interpolating the interior ofthe histogram to produce a PDF on a scale that matches the verticalextent and granularity of the eye type diagrams to be produced.

The method of “morphing” is well known. A form of morphing is preferablyused to transform one fitted histogram to another is applied in order tofill in the space between the N histograms in order to construct acomplete picture over the entire horizontal extent of the eye diagram.Of course, other forms of combination of the various resultinghistograms may be employed. Once the set of PDFs [32] is calculated, toproduce a column for every horizontal coordinate (for every column) ofthe desired PDF eye object [33] a “morphing” procedure is used [26]. Ifa flat-eye is desired, the PDFs may then be offset to have zero mean[27]. Next for each category, each column's PDF is summed [28] into apre-initialized 2D PDF object [33] which is nothing more complicatedthan a two-dimensional array. Next for each column, the PDF may beintegrated or summed to form a CDF.

In accordance with one or more preferred embodiments of the invention,there may be two ways to perform this summing depending on whether thedesired final objects are to be “data centric” or “signal centric”. TheData Centric method sums in such a way as to calculate the probabilitythat the variations from the trajectory encroach or impact the center ofthe eye region (where data values are sampled in a real receiver). So inthis case the probability of encroachment and therefore impact on thecentral region of the eye is highlighted, while little interest in thevariations away from the center of the eye are considered. The signalcentric method calculates the probability of variations away from themean trajectory. It is interesting that both methods produce the sameprobabilities in the 2D CDF for the central region. But the signalcentric version of the 2D CDF is one that contains information outsidethe central region of the eye. Both are interesting and may be used andemployed in accordance with the various embodiments of the presentinvention.

Each of these sums is then summed (according to it's frequency ofoccurrence for the non-repeating case) into the 2D CDF object [34],completing the creation of all three objects of the second step [6] ofFIG. 3.

Because each category of UI is analyzed independently, any ambiguity ofwhether contributions to the 2D CDF are from rising or falling edges iscompletely avoided. Furthermore both the 2D PDF and the 2D CDF are orcan be compensated for the measuring instruments inherent noise.

As more waveforms are added to the procedure, a more precise a fitresults, more accurately reflecting the underlying statistics of theobserved noise. Consequently the estimate of the shape of each onedimensional PDF is convergent, which means the resulting 2D PDF object[33] and 2D CDF object [34] are also both convergent.

Next the third step [7] in FIG. 3, is to create displays from theobjects created in the previous step. To preface these remaining steps,creating “eye-like” views is possible once the 2D PDF and the 2D CDF arecalculated. The first such display is the PDF-eye [52] shown in FIG. 9.This PDF-eye [52] is preferably calculated from the 2D PDF and usingknown display techniques while stopping the translation to color at someminimum probability density (e.g. 1.0e-30 for this particular exemplarycase). This PDF-eye [52] is convergent and would appear virtuallyunchanged for one hundred million UI as it would for one million UI.

From the 2D CDF object [34] there are a number of different displaysthat may be provided. A contour plot may be made of the “signal centric”type as shown in FIG. 10. In this plot each line corresponds to a lineof constant probability of the signal deviating from it's normaltrajectory. The probabilities for 1.0e-6, 1.0 e-7, . . . 1.0e-15 areshown [53] for this particular exemplary embodiment of the invention. Amethod for generating such a plot, including determining the locationsof the lines of constant probability, first an image of the 2D CDFobject [34] is made [55] as shown in FIG. 11. The mostly continuousprobabilities are then “terraced”, whereby for a range of the variableK, each probability that is greater than or equal to 10^(−K), but lessthan 10^(−(K+1)) is reset to the value 10^(−K). Then each “pixel” in thedisplayed image which has a smaller value adjoining it becomes a line ofconstant probability. For the image [54] displayed in a particularexemplary embodiment of the invention, the range of K is 6 to 21 insteps of 1. Sometimes there may be too many lines too close together, sothe steps can be increased as for [53] which shows the lines of constantprobability for 1e-6, 1e-9, 1e-12 and 1e-15 (i.e. K=6, 9, 12 and 15).The most common contour plots in prior-art are displayed as data centriccontours. That is they reflect the absolute probabilities that thesignal under observation approaches the nominal sampling point of adigital receiver near the center of the eye. In FIG. 12 both the contour[56] and the display of lines of constant probability [57] are shown.These probabilities are closely related to the bit error ratio (BER) andso, often the [57] is called the IsoBER plot. A significant by-productof having these data centric contour plots, as is shown in FIG. 13, isobtained by producing a vertical slice noise bathtub curve [59] andhorizontal slice jitter bathtub curve [58] across the data centriccontour plot [56]. The jitter bathtub curve [58] is typically obtainedfrom an analysis of jitter, yet apart from performing CDR to establishthe start and end of each UI, no direct jitter analysis is traditionallyperformed. Using well known methods, in accordance with the variousembodiments of the present invention, the provided inventive data setscan each be analyzed to obtain jitter decomposition for yielding totaljitter (Tj), random jitter (Rj), deterministic jitter (Dj) and totalnoise (Tn), random noise (Rn), deterministic noise (Dn).

Another display flat CDF eye [60] shown in FIG. 14, provided inaccordance with yet another embodiment of the invention, is consideredto be quite unusual by the inventors of the present invention, and istherefore considered to be quite useful and novel. Like all other eyediagrams this image represents behavior of the signal under observationover a region spanning somewhat more than one UI. For illustrativepurposes the flat start of UI [61] and flat end of UI [62] are shown. Asdescribed in the procedure above, the 1-diminsional PDF's have beenoffset to have zero vertical mean. The consequence of this seeminglyarbitrary choice, is to remove the trajectories for each categorycontributing to the 2D PDF object [33] and 2D CDF object [34]. Byremoving the means, or the mean trajectory, the remaining informationshows only noise. This is extremely important for at least tworeasons: 1) the bulges at flat start of UI [61] and flat end of UI [62]indicate a contribution to the measured noise that is caused by jitterand 2) any systematic crosstalk, as might be expected for a nearby datachannel will manifest as a systematic bulge in this display-type. Sothis novel display type is useful to identify both jitter and crosstalk.

Another line of analysis is shown in FIG. 15 and leverages the databaseof means by category [31] as well as the expected transition times [19]and [20] for the most recent [17] (and any number of subsequentacquisitions). For each data point of each input waveform [63] UI in theinput waveform [63], the [20] may first be used to obtain a category.The database of means by category [31] for that category may then befurther employed in conjunction with the expected transition times [19]to define the start and end of the UI. As a result, a new data point maybe calculated for every data point in the input waveform [63] (with theexception of some points at the beginning and some at the end of thewaveform lacking adequate surrounding points to identify a category) byinterpolating from the database of means by category [31] for thecategory points at the same horizontal (time) coordinate. In this way, anew systematic waveform [64] is preferably created from which noise andjitter have been substantially removed. Subtracting this systematicwaveform [64] from the input waveform [63] produces a residual waveform[65] which consists of only the non-systematic part of the waveformunder analysis. This waveform can be analyzed to obtain spectrum of theresidual waveform [66] of only the non-systematic part of the inputwaveform, which is a critical.

It should also be understood that the invention, while describedgenerally as a set of method steps and resulting images, is equallyapplicable to a computer program stored to a non-transitory mediumwhich, when run on a general purpose computer and processor, or otherspecialized hardware, such as an oscilloscope or other test andmeasurement apparatus, and including one or more of an acquisitionchannel, clock recovery module, processor, memory communication systemand the like to support operation of the computer program. Therefore,the invention is equally applicable to a hardware system including aprocessor for allowing the system to perform the desired method steps.Furthermore, this system may be positioned locally, remotely, spreadover multiple locations, and may include cloud or other remote computingsystems and/or storage.

It will thus be seen that the objects set forth above, among those madeapparent from the preceding description, are efficiently attained and,because certain changes may be made in carrying out the above method andin the construction(s) set forth without departing from the spirit andscope of the invention, it is intended that all matter contained in theabove description and shown in the accompanying drawings shall beinterpreted as illustrative and not in a limiting sense.

It is also to be understood that the following claims are intended tocover all of the generic and specific features of the invention hereindescribed and all statements of the scope of the invention which, as amatter of language, might be said to fall therebetween.

What is claimed is:
 1. A method, comprising: acquiring, by an electronicmeasurement instrument, an input waveform; sampling, by the electronicmeasurement instrument, the input waveform to identify values of theinput waveform at different times; identifying, by the electronicmeasurement instrument, multiple instances of a recurring portion of theinput waveform; identifying, by the electronic measurement instrument,multiple values for a first point in time within the same multipleinstances of the recurring portion of the input waveform, wherein themultiple values for the first point in time were gathered by selecting avalue for the first point in time from each of the same multipleinstances of the recurring portion of the input waveform; generating, bythe electronic test measurement, a first mathematical equation thatrepresents a distribution of the multiple values for the first point intime within the same multiple instances of the recurring portion of theinput waveform; identifying, by the electronic measurement instrument,multiple values for a second point in time within the same multipleinstances of the recurring portion of the input waveform, wherein themultiple values for the second point in time were gathered by selectinga value for the second point in time from each of the same multipleinstances of the recurring portion of the input waveform; generating, bythe electronic test measurement, a second mathematical equation thatrepresents a distribution of the multiple values for the second point intime within the same multiple instances of the recurring portion of theinput waveform; and displaying, by the electronic measurementinstrument, a presentation of the recurring portion of the inputwaveform, including: (i) using the first mathematical equation torepresent the distribution of the multiple values for the first point intime within the same multiple instances of the recurring portion of theinput waveform, and (ii) using the second mathematical equation torepresent the distribution of the multiple values for the second pointin time within the same multiple instances of the recurring portion ofthe input waveform.
 2. The method of claim 1, wherein displaying thepresentation of the recurring portion of the input waveform includespresenting an eye diagram of the recurring portion of the inputwaveform.
 3. The method of claim 1, wherein generating the firstmathematical equation that represents the distribution of the multiplevalues for the first point in time includes generating a histogram ofthe multiple values for the first point in time and creating the firstmathematical equation to represent the histogram.
 4. The method of claim1, wherein identifying the multiple values for the first point in timeincludes interpolating the values of the input waveform that wereidentified by the sampling.
 5. The method of claim 1, whereinidentifying the recurring portion of the input waveform includesperforming a clock data recovery process that analyzes the inputwaveform to determine one or more expected transition times in the inputwaveform.
 6. The method of claim 1, wherein the recurring portion of theinput waveform represents multiple unit intervals that are similar. 7.The method of claim 1, further comprising: identifying, by theelectronic measurement instrument, a plurality of unit intervals; andcategorizing, by the electronic measurement instrument, multiple unitintervals from among the plurality of unit intervals as having a similaror identical history, wherein the recurring portion of the inputwaveform represents the multiple unit intervals.
 8. The method of claim3, wherein creating the first mathematical equation to represent thehistogram includes interpolating the histogram to obtain the firstmathematical equation.
 9. An electronic test instrument, comprising: aninput to acquire an electronic waveform; a processor; non-transitorymedium storing a computer program that, when executed by the processor,causes the electronic test instrument to: (i) sample the input waveformto identify values of the input waveform at different times, (ii)identify multiple instances of a recurring portion of the inputwaveform, (iii) identify multiple values for a first point in timewithin the same multiple instances of the recurring portion of the inputwaveform, wherein the multiple values for the first point in time weregathered by selecting a value for the first point in time from each ofthe same multiple instances of the recurring portion of the inputwaveform, (iv) generate a first mathematical equation that represents adistribution of the multiple values for the first point in time withinthe same multiple instances of the recurring portion of the inputwaveform, (v) identify multiple values for a second point in time withinthe same multiple instances of the recurring portion of the inputwaveform, wherein the multiple values for the second point in time weregathered by selecting a value for the second point in time from each ofthe same multiple instances of the recurring portion of the inputwaveform, and (vi) generate a second mathematical equation thatrepresents a distribution of the multiple values for the second point intime within the same multiple instances of the recurring portion of theinput waveform; and a display to present the recurring portion of theinput waveform, including: (i) using the first mathematical equation torepresent the distribution of the multiple values for the first point intime within the same multiple instances of the recurring portion of theinput waveform, and (ii) using the second mathematical equation torepresent the distribution of the multiple values for the second pointin time within the same multiple instances of the recurring portion ofthe input waveform.
 10. The electronic test instrument of claim 9,wherein the presentation of the recurring portion of the input waveformincludes a presentation of an eye diagram of the recurring portion ofthe input waveform.
 11. The electronic test instrument of claim 9,wherein generating the first mathematical equation that represents thedistribution of the multiple values for the first point in time includesgenerating a histogram of the multiple values for the first point intime and creating the first mathematical equation to represent thehistogram.
 12. The electronic test instrument of claim 9, whereinidentifying the multiple values for the first point in time includesinterpolating the values of the input waveform that were identified bythe sampling.
 13. The electronic test instrument of claim 9, whereinidentifying the recurring portion of the input waveform includesperforming a clock data recovery process that analyzes the inputwaveform to determine one or more expected transition times in the inputwaveform.
 14. The electronic test instrument of claim 9, wherein therecurring portion of the input waveform represents multiple unitintervals that are similar.
 15. The electronic test instrument of claim9, wherein the stored computer program, when executed, causes theelectronic test instrument to: identify a plurality of unit intervals;and categorize multiple unit intervals from among the plurality of unitintervals as having a similar or identical history, wherein therecurring portion of the input waveform represents the multiple unitintervals.
 16. The electronic test instrument of claim 11, whereincreating the first mathematical equation to represent the histogramincludes interpolating the histogram to obtain the first mathematicalequation.